What is the dimensional formula of gravitational constant ?

Options

A. `[M^(-1)L^(3)T^(-2)]`
B. `[M^(-1)L^(3)T^(-1)]`
C. `[M^(-2)L^(3)T^(-2)]`
D. `[M^(-2)L^(-1)T^(3)]`

Answer

To find the dimensional formula of the gravitational constant \( G \), we can start from Newton's law of gravitation, which states:

\[ F = \frac{G m_1 m_2}{r^2} \]

Where:
- \( F \) is the gravitational force,
- \( G \) is the gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses of the two objects,
- \( r \) is the distance between the centers of the two masses.

### Step 1: Rearranging the formula for \( G \)

We can rearrange the formula to solve for \( G \):

\[ G = \frac{F r^2}{m_1 m_2} \]

### Step 2: Identifying the dimensions of each variable

Next, we need to identify the dimensional formulas of the variables involved:

- The dimensional formula for force \( F \) is given by:
\[ [F] = [M L T^{-2}] \]

- The dimensional formula for distance \( r \) is:
\[ [r] = [L] \]

- The dimensional formula for mass \( m_1 \) and \( m_2 \) is:
\[ [m_1] = [m_2] = [M] \]

### Step 3: Substituting the dimensions into the equation for \( G \)

Now, substituting the dimensional formulas into the equation for \( G \):

\[ G = \frac{[M L T^{-2}] \cdot [L^2]}{[M] \cdot [M]} \]

### Step 4: Simplifying the expression

Now, we can simplify the expression:

\[ G = \frac{[M L T^{-2}] \cdot [L^2]}{[M^2]} = \frac{[M L^3 T^{-2}]}{[M^2]} \]

This simplifies to:

\[ G = [M^{-1} L^3 T^{-2}] \]

### Final Result

Thus, the dimensional formula of the gravitational constant \( G \) is:

\[ [G] = [M^{-1} L^3 T^{-2}] \]